| L(s) = 1 | − 4·3-s − 8·7-s + 8·9-s − 8·13-s − 8·17-s + 20·19-s + 32·21-s + 8·23-s − 8·27-s + 24·37-s + 32·39-s + 20·41-s + 8·43-s + 32·49-s + 32·51-s + 16·53-s − 80·57-s + 8·59-s − 16·61-s − 64·63-s − 12·67-s − 32·69-s + 8·73-s + 16·79-s − 7·81-s − 4·83-s + 64·91-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3.02·7-s + 8/3·9-s − 2.21·13-s − 1.94·17-s + 4.58·19-s + 6.98·21-s + 1.66·23-s − 1.53·27-s + 3.94·37-s + 5.12·39-s + 3.12·41-s + 1.21·43-s + 32/7·49-s + 4.48·51-s + 2.19·53-s − 10.5·57-s + 1.04·59-s − 2.04·61-s − 8.06·63-s − 1.46·67-s − 3.85·69-s + 0.936·73-s + 1.80·79-s − 7/9·81-s − 0.439·83-s + 6.70·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.192474866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.192474866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 + 4 T + 8 T^{2} + 8 T^{3} + 7 T^{4} + 8 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 443 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 72 T^{3} + 146 T^{4} + 72 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 176 T^{3} + 943 T^{4} + 176 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 152 T^{3} + 706 T^{4} - 152 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 2198 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2694 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 24 T^{3} - 1582 T^{4} - 24 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 4222 T^{4} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1168 T^{3} + 10258 T^{4} - 1168 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 984 T^{3} + 13223 T^{4} + 984 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 432 T^{3} + 5471 T^{4} - 432 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 232 T^{3} + 6103 T^{4} + 232 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 210 T^{2} + 22163 T^{4} - 210 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 2498 T^{4} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.79301250209711840640584163462, −6.25587759262286638103069358365, −6.15442653768003564207914336442, −6.08987986884258970862303418479, −6.05397273512797337359265838590, −5.48803813828730591182458181258, −5.45913186955469515957402545657, −5.37602011770929950985856289247, −5.20223251348097933706948156460, −4.57369418392983996212544053728, −4.51738884655862349558140498924, −4.48717350748919592025804429003, −4.22168954388882450686365735309, −3.91689602141406857303457998534, −3.29892563119849109391621122070, −3.27833638958792849593598535698, −2.89901667049167228520355124873, −2.80395355190639347462902821278, −2.74300513435413772548713468376, −2.26011762162480818249902262632, −1.95591485598904206856418492912, −1.05993373866904518575775006589, −0.76892180741799599700589028155, −0.66394515891896171358083144374, −0.53714113357900642819312602688,
0.53714113357900642819312602688, 0.66394515891896171358083144374, 0.76892180741799599700589028155, 1.05993373866904518575775006589, 1.95591485598904206856418492912, 2.26011762162480818249902262632, 2.74300513435413772548713468376, 2.80395355190639347462902821278, 2.89901667049167228520355124873, 3.27833638958792849593598535698, 3.29892563119849109391621122070, 3.91689602141406857303457998534, 4.22168954388882450686365735309, 4.48717350748919592025804429003, 4.51738884655862349558140498924, 4.57369418392983996212544053728, 5.20223251348097933706948156460, 5.37602011770929950985856289247, 5.45913186955469515957402545657, 5.48803813828730591182458181258, 6.05397273512797337359265838590, 6.08987986884258970862303418479, 6.15442653768003564207914336442, 6.25587759262286638103069358365, 6.79301250209711840640584163462
